Derive Birkhoff Ergodic Theorem from Von Neumann

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wrobel
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Is that possible to derive the Birkhoff Ergodic Theorem from or with the help of the Von Neumann ergodic theorem?
 
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It seems to come down to ( I just read about it now)to showing that a.e convergence implies conbegence in mean square. Is this correct?
 

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